Effective Fluctuation and Response Theory

The so-called fluctuation theorems pushed the study of systems far beyond equilibrium, whose response to thermodynamic forces (affinities) is characterized by the reciprocal and the fluctuation-dissipation relations. All these results rely on the assumption that the observer has complete information about the system: no hidden leakage to the environment, exact evaluation of the thermodynamic cost of processes. Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only be able to establish local equilibrium, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations? Within the formalism of Markov jump processes on finite networks, we provide a broad theory of the statistical behavior of some out of many currents that flow across a system. In particular (1) There exist effective affinities for which an integral fluctuation relation holds; (2) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds; (3) Under reasonable assumptions on the parametrization of the rates, effective affinities can be operationally defined by a procedure of tuning to stalling; (4) There exists a notion of marginally time-reversed process which restores the full-fledged fluctuation relation and reciprocity; (5) There exist fluctuation relations across different levels in the hierarchy of more and more “complete” theories. The above results apply to configuration-space currents, and to their phenomenological linear combinations provided certain symmetries of the effective affinities are respected—a condition whose range of validity we deem the most interesting question left open to future inquiry.